226
5 Formulation and Implementation of the Gamow Shell Model
the configurations |i B for all possible number of particles n B and total angular
momentum j B are constructed. This augments the size of the B space and generates
a set of basis states (|k A (n A , j A ) ⊗ |i B (n B , j B ))
J . n A(B) and j A(B) are respectively
the numbers of particles and angular momenta of the k A (i B ) states and J is the total
angular momentum of the calculated eigenstate.
When the number of configurations i B of the B space reaches N opt , of the order
of 10 to 100 typically, the Hamiltonian is diagonalized to provide an approximation
of the calculated nuclear state:
|Ψ J =
k A ,i B
c
k A
i B
{|k A ⊗ |i B
J .
(5.72)
The latter has been determined by the overlap method (see Sect. 5.4) [21]. The
reduced density matrix ˆ
ρ B
i B i
B
is thus calculated from Eq. (5.72):
ˆ
ρ
j B
i B i
B
=
k A
c
k A
i B
c
k A
i
B
,
(5.73)
where the angular momentum j B , which is the same for |i B and |i B , is fixed. The
reduced density matrix is then diagonalized, and the resulting eigenvectors which
possess the largest eigenvalues in modulus (at most N opt of them) are kept in the
model space, the other ones being rejected. This method is indeed motivated by the
variational principle, which states that the wave function bearing the largest overlap
with the exact nuclear state is the one built from the aforementioned eigenvectors
[40, 41]. The eigenvectors then generate the new B space, and the sub-operators
associated to the Hamiltonian are recalculated in this new space.
When all the shells of the Berggren basis have been taken into account, the
“warm-up” phase terminates and the “sweep” phases start [38]. They are similar
to the “warm-up” phase, except that the states of the space A are now replaced
by the (|k A (n A , j A ) ⊗ |i B (n B , j B ))
J states, issued from the previous iteration.
The scattering states (n, ,, j) of the Berggren basis are then reintroduced one by
one, with the index associated to (n, ,, j) firstly decreasing (“sweep down”) and
afterward increasing (“sweep up”) [38,39]. Indeed, the truncation applied after each
diagonalization of the reduced density matrix suppresses correlations, which must
be recovered during sweeps. Convergence is obtained after a few “sweeps” [38, 39].
Everytime a new scattering shell is added during a sweep, which translates into
diagonalization of new reduced density matrice and recalculation of Hamiltonian
sub-operators, by definition, a “step” occurs. The speed of convergence is measured
via their number N step .
The density matrix renormalization group approach has been tested by calculating the 3/2
−
1 ground state and the 1/2
−
1 first excited state of 7 He [38] and the ground
states of the 7 Li and 8 Li nuclei [39], using the same Hamiltonian as in Sect. 5.4.
The rapid convergence of the density matrix renormalization group approach is
illustrated in Fig. 5.5 for the 7 He and 7 Li nuclei.
5 Formulation and Implementation of the Gamow Shell Model
the configurations |i B for all possible number of particles n B and total angular
momentum j B are constructed. This augments the size of the B space and generates
a set of basis states (|k A (n A , j A ) ⊗ |i B (n B , j B ))
J . n A(B) and j A(B) are respectively
the numbers of particles and angular momenta of the k A (i B ) states and J is the total
angular momentum of the calculated eigenstate.
When the number of configurations i B of the B space reaches N opt , of the order
of 10 to 100 typically, the Hamiltonian is diagonalized to provide an approximation
of the calculated nuclear state:
|Ψ J =
k A ,i B
c
k A
i B
{|k A ⊗ |i B
J .
(5.72)
The latter has been determined by the overlap method (see Sect. 5.4) [21]. The
reduced density matrix ˆ
ρ B
i B i
B
is thus calculated from Eq. (5.72):
ˆ
ρ
j B
i B i
B
=
k A
c
k A
i B
c
k A
i
B
,
(5.73)
where the angular momentum j B , which is the same for |i B and |i B , is fixed. The
reduced density matrix is then diagonalized, and the resulting eigenvectors which
possess the largest eigenvalues in modulus (at most N opt of them) are kept in the
model space, the other ones being rejected. This method is indeed motivated by the
variational principle, which states that the wave function bearing the largest overlap
with the exact nuclear state is the one built from the aforementioned eigenvectors
[40, 41]. The eigenvectors then generate the new B space, and the sub-operators
associated to the Hamiltonian are recalculated in this new space.
When all the shells of the Berggren basis have been taken into account, the
“warm-up” phase terminates and the “sweep” phases start [38]. They are similar
to the “warm-up” phase, except that the states of the space A are now replaced
by the (|k A (n A , j A ) ⊗ |i B (n B , j B ))
J states, issued from the previous iteration.
The scattering states (n, ,, j) of the Berggren basis are then reintroduced one by
one, with the index associated to (n, ,, j) firstly decreasing (“sweep down”) and
afterward increasing (“sweep up”) [38,39]. Indeed, the truncation applied after each
diagonalization of the reduced density matrix suppresses correlations, which must
be recovered during sweeps. Convergence is obtained after a few “sweeps” [38, 39].
Everytime a new scattering shell is added during a sweep, which translates into
diagonalization of new reduced density matrice and recalculation of Hamiltonian
sub-operators, by definition, a “step” occurs. The speed of convergence is measured
via their number N step .
The density matrix renormalization group approach has been tested by calculating the 3/2
−
1 ground state and the 1/2
−
1 first excited state of 7 He [38] and the ground
states of the 7 Li and 8 Li nuclei [39], using the same Hamiltonian as in Sect. 5.4.
The rapid convergence of the density matrix renormalization group approach is
illustrated in Fig. 5.5 for the 7 He and 7 Li nuclei.